Mapping vs. Rounding (was: [MMM] Short piece using tempered 2.3.7/5.11/9 tonality diamond)

I am responding here to make sure that people who are interested on MMM see the answer, because I think it's very important. I'm also cc'ing the tuning list, because I think any further discussion should take place there.

Mike said:
> I think I had this argument with Gene about patent vals ages
> ago when designing a program to find optimum scales under 31EDO...
> And this is a shining example of the difference between how I
> think notes should be remapped in both re-tuning and tuning theory
> in general (closest error) vs. how patent vals map them.

To clarify one thing: The patent val may not always be the val you want; in the current "shining example" case, for example -- which is really a best case for you, rather than a good example generally -- changing one number in the val will give you a better mapping for this specific scale.

Respectfully, though, I think you're wrong to only focus on the error of a individual intervals. Whatever makes you happy is fine, of course, but you're losing more than you're gaining.

Here's why:

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1. Using a subset of an EDO to represent a JI scale always introduces error. Error isn't always bad, of course; it's this error that lets us temper out a specific comma, for instance. How that error gets applied, however, is critical.

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2. When you round intervals to their nearest EDO steps, you try to minimize the error for each interval independently. Because you're considering each interval independently, there's no mathematical relationship between the number of EDO steps for one interval vs. another. In other words, this isn't a regular temperament.

As a result, "31 EDO tempers out 352/351" isn't true for your scale. It can't be, because the primes 2, 3, 11, and 13 aren't getting mapped to the EDO. In fact, the notion of commas becomes *meaningless*. If you do get relationships that look like an interval has been tempered out, they're *coincidental* -- but *not* a necessary result of the process.

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3. When you use a formal mapping to determine which EDO steps represent a JI interval, you are applying the error in a consistent way.

When given a collection of monzos that represent your intervals, you use the same val for the EDO to create homomorphisms. (In other words, the single val <V| is used with each interval's monzo |M> to create <V|M>, which tells you the number of EDO steps you should use to represent each interval.)

Because these homomorphisms are based on the primes of the intervals AND the way these primes get mapped to the EDO, this *is* a regular temperament. You *can* say "31 EDO tempers out 352/351" because that's a necessary result of using the mapping process to get the EDO steps for each interval.

This sounds complex, but it's really not. I have a simple spreadsheet that does the job for me.

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In the current case, you had only one 19-limit interval in the scale. Because of that, we could adjust the 19-limit val without affecting any other intervals. It's therefore a best-possible-case scenario for the "rounding" approach vs. the "mapping" approach, because the rounding doesn't really do any damage to the relationships between the EDO intervals, and you ended up with a scale that *could* result from a val; however, that's *coincidental*, not *necessary*. On the other hand, if you had had other 19-limit intervals, choosing the rounded number for one versus the mapped number for another would have resulted in having an irregular temperament, and all notions of "such-and-such comma is tempered out" would have flown out the window.

You may not like the fact that regular temperaments can give you the "wrong" interval, i.e., one that you wouldn't have picked if you were rounding. I didn't either, which is why I thought I was doing something wrong -- but I wasn't. As Graham succinctly explained, "When you choose a regular temperament, you give an inherent error to each prime interval (mapping of a prime number). Those errors accumulate when you look at smaller intervals. After a point, you'll always [have] something with a higher error than the step size you're dealing with." Perfectly said.

Thus using mappings (i.e., vals) to create a scale has a significant theoretical advantage over rounding for the general case. It *is* Paul Erlich's "Middle Path" between JI and EDOs, whereas rounding is not. If you like things like Porcupine and Mavila, you need to pay attention to the mappings, and not just rounding.

Obviously, you don't have to follow the theory, and you can use any theory (or no theory!) to create scales that make you happy. Also, if you have a physical instrument that can't step outside of a certain EDO, you may use what you can despite the fact that it's not theoretically the right way to go. But the mapping theory is clearly superior in most cases to the rounding theory.

To sum up: If you care about tempering commas, you should use the mapping method. Also, it seems to me that the mapping method is an important part of tuning theory because it retains the relationships of the primes in the intervals, whereas rounding is not because it does not.

Regards,
Jake